Saturday, November 1

IP7
Lifting the Curse on the Hamiltonian and Symplectic Eigenproblem

A new method is presented for the computation of the generalized eigenvalues and the stable invariant subspaces of real Hamiltonian or symplectic pencils and matrices. Simultaneously this method can be used to obtain the stabilizing solutions of the continuous-time and discrete-time algebraic Riccati equation. These problems are of great importance in the solution of optimal control and H-infinity control problems.

The new method that we propose is numerically backwards stable, has complexity O(n^3) and uses the Hamiltonian or symplectic structure to a maximum extend and in this sense solves the open problem known as Van Loan's curse.

The main ingredients for the new approach are the relationship between the invariant subspaces of a Hamiltonian matrix H and the extended matrix \mat{cc}0&H\\ H&0\rix and the symplectic URV-decomposition. (This is joint work with P. Benner and H. Xu.)